Periodic solutions to nonlinear parabolic differential equations
Robert E. Gaines, Wolfgang L. Walter · Rocky Mountain Journal of Mathematics · 1977
Introduction.We consider periodic solutions of the nonlinear parabolic equation ( 1)Some of our results apply only to the special case of the quasilinear equation ( 3)In § 1 we give general sufficient conditions for the existence of a priori bounds on T-periodic solutions u(t, x) to ( 1) -(2).In § 2 we obtain bounds on derivatives of T-periodic solutions to (3) -(2).In § 3 we use degree theory in conjunction with results of Fife [2] for the linear case to obtain existence of T-periodic solutions to (3) -(2).Periodic solutions of nonlinear parabolic equations have been studied by Bange [1], Fife [2], Prodi [6], Vaghi [7], Walter [9] and others.The results presented here apply to a wider class of equations with one space variable than those presented previously.Our a priori bound arguments do not require semi-linearity, monotonicity, or Lipschitz conditions of the nonlinear functions involved as in [2], [7], and [9], and do not require complicated analysis of Green's function representations as in [1] and [6]. Bounds for u(t 9 x).We first present a general theorem which generalizes and simplifies the procedure developed by one of the authors (see [4] ) for obtaining a priori bounds on solutions to nonlinear second order ordinary differential equations.In this section we assume:(i) f:RX (0,1) X fl 3 -* R, (ii) /(*, x, z, p, fi) ^ f(t, x, z, p, r 2 ) for r x è r 2 , and (iii) f(t + T, x, z, p, r) = f(t, x, z 9 p, r) (T-periodic).A solution will be a function u(t, x) which is continuous on R X [0,1]